Earthquakes from the point of view of mechanics of contact interaction of surfaces

. The work is devoted to earthquake forecasting. The physics of an earthquake, despite the fact that this problem has been facing humanity for more than 100 years, it still remains unclear. What is the reason and why this is a very important problem, which is being solved by a huge number of universities, institutes and companies around the world. The physical model of a tectonic earthquake is presented in the form of a plate connection loaded with a shear force, in which the non-displacement is provided by friction forces. In calculating the contact interaction, the pliability of the plates and the tangential pliability of the contact layer are taken into account. A system of equations has been obtained, the solution of which makes it possible to specify to some extent the location, strength and time of the earthquake occurrence.


Introduction
Despite modern technical and technological progress, the expansion of networks for recording seismic activity and a huge amount of accumulated data on the phenomenological patterns of earthquakes, this phenomenon remains one of the most catastrophic and insufficiently predicted natural disasters. The importance of studying the nature of earthquakes increases due to the increasing population in the vicinity of large active faults.
An earthquake is the result of slippage on a pre-existing geological fault. Geologists have known for a long time that any part of the earth's crust, both on the surface and at considerable depths, contains a huge number of in homogeneities and defects of various scales from micro-and nanostructures to transcontinental faults [1][2][3].
The resolution of modern seismological methods makes it possible to not only measure the average fracture displacement and stress relief, but also to estimate their distribution both in time and in space. The variation of these parameters within a single rupture turns out to be very significant, which is undoubtedly due to the heterogeneity of the structure, frictional properties and stress level on the fault plane.
As is known, 95% of earthquakes are tectonic earthquakes caused by the inconsistency of the movements of the tectonic plates of the earth's pores. Tangential stresses arise along the contact surface of these plates. If the tangential stresses exceed the specific frictional forces, contact failure occurs in the contact layer. Then the rock layers in the contact zone shift sharply, emitting seismic waves.
When constructing concrete structures, as well as nuclear power plants, it is important to take into account the seismological situation of the area in order to destroy them [4,5].
Historically, the development of approaches to the description of mechanisms for the preparation of an earthquake was due to the great desire of mankind to learn how to predict this phenomenon, entailing huge human and economic losses. This branch of science received the most intensive development in the second half of the last century, which was caused by the emergence and development of complex networks of continuous seismic monitoring. The continuous operation of such networks, coupled with geophysical and geodetic observations in a number of earthquake-prone areas, made it annoying to begin to receive orders of magnitude more information about the processes preceding earthquakes and their accompanying ones.
According to modern concepts, it is considered established that an earthquake is a mechanical process of destruction of rocks of the earth's crust with the release of energy accumulated in the edges of lithospheric plates in a long complex process of increasing deformation during its preparation. Therefore, when mathematically modelling the earthquake process, it is necessary to proceed from the basic equations of continuum mechanics.
For the first time, the idea of the earthquake process as a discontinuity was clearly noted by Harry Reed in 1910, after a thorough geodetic study of movements along the San Andreas fault during the San Francisco earthquake of 1906. In the future, this concept of the occurrence of an earthquake was called the theory of "elastic recoil". Reed based his theory on the following premises: -an earthquake occurs at the moment when the lithosphere splits along the surface, which can be defined as a geological fault; -an earthquake is preceded by a gradual increase in elastic stresses on both sides of the fault; -during an earthquake, both sides of the fault mutually move by an amount that exactly corresponds to the complete removal of elastic stresses along the fault.
The essence of the Raid theory consists in the accumulation of elastic deformations with a gradual increase in the movement of blocks, the formation of a gap and a sharp shift of the sides of the gap to a position in which there are no elastic deformations. A rupture, a crack, the actual focus of an earthquake, according to the Raid, can come to the surfaceand then we either observe a strong earthquake, or be under it -in all cases of weak earthquakes.
At the moment, there are only a few successful forecasts of major earthquakes issued over the past 60-70 years, while strong earthquakes capable of causing extensive destruction occur on the planet about once every two weeks. This state of affairs led to the fact that at the turn of the century, optimism in the search for predictive signs in geophysical and seismological fields was replaced by deep pessimism. There have been publications claiming that earthquakes are purely random and a deterministic approach is hopeless, and as a result, earthquake prediction is fundamentally impossible. Since the mid-90s of the last century, practically no new ideas have appeared in the problem of earthquake forecasting in the West. The negative result regarding the forecast allowed us to come to the understanding that our ideas about the mechanism of earthquake generation are quite far from the real natural process and it is necessary to return to rethinking the root cause of the earthquake -the mechanisms of deformation of large rock masses.
A comprehensive study of the structure of the geomedium in earthquake-prone areas, taking into account the tectonic, lithological and fluid heterogeneity of its structure, will provide the necessary information to understand the processes of formation and development of earthquake preparation areas, including their spatial and temporal localization.
The modern knowledge obtained within the framework of the designated areas will allow us to formulate a new model of earthquake preparation, which is currently missing, and the need for it is obvious.
Therefore, the most important question of seismology is to clarify the nature of the earthquake mechanism, i.e. the physics of processes in its focus.

Main Part
The geological environment is a block-connected structure with a wide range of element sizes, from micro-scales to the scales of regions and entire plates. The parameters of the structure and the variability of various fields in them are due to various processes occurring over significant areas and volumes, both laterally and in depth. Naturally, for processes developing in local zones during this period, external influences may be phenomena occurring in neighbouring local zones, for example, elastic waves of seismic events of different strengths. In many works, it is stated that at various large-scale levels of the lithosphere and mantle there is a continuous process of reorganization of the structure and that each level affects adjacent ones, thereby creating a complex nature of physico-chemical fields reflecting these processes in the complex. The consequence of these processes is the manifestation of a tectonic flow, including the movement of the masses of the rock environment based on plastic, solid-plastic and discontinuous phenomena. This reorganization of the structure of the various shells of the Earth is associated with its endogenous activity initiated and supported by processes in the core and on its boundary with the mantle. In the upper mantle and lithosphere, endogenous activity is most noticeably realized in boundary or discontinuous structures (boundaries of blocks, plates, rifts, structure in homogeneities, etc.). Endogenous activity in geological representations is carried out by fluid and fluid-magmatic flows. The same flows can influence the evolution of block structures.
The movement of blocks in the intraplate space also raises questions about the place of application of the acting forces. Previously, it was believed that the consolidated crust, once formed, is not subject to serious structural transformations in the future and the position of various boundaries does not change in it. Recent studies show that this is not the case. Over time, there is a structural and material transformation of the primary crust, as well as an increase in the granite-metamorphic layer. The formation of new volumes of granitemetamorphic layers and changes in their thickness and parameters are associated with endogenous and exogenous processes.
The tectonic heterogeneity of the medium determines its block structure. Seismic deformations are localized in weakened areas of the medium, which include interblock boundaries and fault systems of various ranks. The block structure of the geomedium affects the nature of its deformation in the field of tectonic stresses and the features of anomalous changes in the parameters of geophysical fields caused by the formation of an earthquake focus.
A classic example of contact interaction is the wheel-rail contact pair, in which the stresses in the contact area and its transmission properties are of the greatest interest. Contact pairs can serve to transfer mechanical energy, electric current and heat, or to prevent the transport of undesirable substances (seals). Examples of contact interaction are both the interaction of the tip of an atomic force microscope with the scanned material, and phenomena in the area of contact between tectonic plates.
A physical model of such phenomena in the first approximation can serve as a pressure connection of p plates ( fig. 1) loaded with a shear force. In the mechanics of contact interaction, it has been theoretically and experimentally established that when the contact zone is squeezed by pressure p, the convergence δ in the contact layer (under repeated loads) is proportional to the arithmetic mean height Ra of the irregularities of the contacting surfaces, increases with pressure and decreases with an increase in the elastic modulus E of the contacting materials where c0 is a dimensionless parameter (for metals and plastics c0 ≈ 500, no studies have been conducted to determine c0 of terrestrial rocks); ε is the scale factor, which at L > 50 mm is greater than one and increases with increasing deviation from flatness, that is, with increasing contact surface.
In this case, the coefficient of contact compliance of the contact layer at pressure p0 is numerically equal to the tangent of the angle α of the slope of the tangent to the graph of the dependence of the convergence δ on pressure at p = p0 When the layer is loaded with a tangential stress τ, an elastic displacement occurs in the contact layer where kτ is the coefficient of tangential compliance, which is approximately equal to the value of the coefficient of contact compliance in the normal direction to the contact surface.
Thus, in practical calculations, the contact layer can be considered as a third body of negligibly small thickness with known compliance in the normal to the contact surface and tangential directions.
The earthquake force, in the first approximation, can be assumed proportional to the displacement amplitude δ when the contacting surfaces slip and the pressure p in contact.
If normal (to the contact surface) or tangential elastic deformations occur at the joint, then free oscillations may occur, respectively, with the frequency of oscillations of normal va or tangential vτ where ja = BL/(kε) is the contact stiffness; m is the oscillating mass; k is the coefficient of contact compliance of the contact layer.
When a vibration with the specified frequency is applied to the joint, the force F, which causes a breakdown of the elastic displacement over the entire contact surface, can significantly decrease.

Conclusions
The strength of a tectonic earthquake depends on which slips occur in the contact layer of tectonic plates: local or over the entire contact surface. The shear force, which does not lead to the appearance of local slip zones, is significantly less than the shear force, which entails the disruption of elastic displacement over the entire contact surface.
Local slippage is noted at the edges of the contact surface. In this case, an earthquake occurs with a force limited in magnitude and decreasing with increasing pressure in contact.
The breakdown of elastic displacement over the entire contact surface entails an earthquake of many times greater force. The harbinger of such a breakdown is the appearance of local slippage in contact.
When vibration with a frequency close to va is applied to the contact area, the force causing the elastic displacement to break down over the entire contact surface may decrease significantly.
It is likely that the vibrations of the Earth's crust during earthquakes occur with the frequencies va and vt.